Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nelbrnel Structured version   Visualization version   GIF version

Theorem nelbrnel 44260
 Description: A set is related to another set by the negated membership relation iff it is not a member of the other set. (Contributed by AV, 26-Dec-2021.)
Assertion
Ref Expression
nelbrnel ((𝐴𝑉𝐵𝑊) → (𝐴 _∉ 𝐵𝐴𝐵))

Proof of Theorem nelbrnel
StepHypRef Expression
1 nelbr 44258 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴 _∉ 𝐵 ↔ ¬ 𝐴𝐵))
2 df-nel 3057 . 2 (𝐴𝐵 ↔ ¬ 𝐴𝐵)
31, 2bitr4di 292 1 ((𝐴𝑉𝐵𝑊) → (𝐴 _∉ 𝐵𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2112   ∉ wnel 3056   class class class wbr 5037   _∉ cnelbr 44255 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2730  ax-sep 5174  ax-nul 5181  ax-pr 5303 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nel 3057  df-v 3412  df-dif 3864  df-un 3866  df-nul 4229  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-br 5038  df-opab 5100  df-nelbr 44256 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator